What is the probability Tim guesses the code correctly first time?
Understand the Problem
The question is asking to calculate the probability that Tim can guess a 4-digit code correctly on his first try, given that he knows certain details about the code, such as its structure and the starting digit.
Answer
The probability is $P = \frac{1}{500}$.
Answer for screen readers
The probability that Tim guesses the code correctly on his first try is $P = \frac{1}{500}$.
Steps to Solve
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Identify the structure of the code
Since the code is a 4-digit code, we know the first digit is fixed as 6 (from the details given). So the code structure is: $6XXX$ where $X$ represents unknown digits. -
Determine valid options for odd digits
The last digit of the code must be odd. The odd digits available are: 1, 3, 5, 7, and 9. This gives us 5 choices for the last digit. -
Count the possibilities for the second and third digits
The second and third digits can be any number from 0 to 9. Thus, each has 10 possible choices. -
Calculate total combinations
The total combinations for the 4-digit code can be calculated by multiplying the possibilities:
- Choices for 2nd digit: 10
- Choices for 3rd digit: 10
- Choices for last digit (odd): 5
Total combinations = $10 \times 10 \times 5 = 500$
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Calculate probability of guessing correctly
There is only one correct code. The probability of guessing it correctly is:
$$ P = \frac{1}{\text{Total combinations}} = \frac{1}{500} $$
The probability that Tim guesses the code correctly on his first try is $P = \frac{1}{500}$.
More Information
This probability indicates that Tim has a small chance of guessing the code correctly, reflecting the combinatorial nature of guessing codes with specific details known.
Tips
- Forgetting that the last digit must be odd, leading to incorrect total possibilities.
- Not realizing that the first digit is already fixed, which simplifies the calculations.
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